The most crucial step of the algorithm is to sample \(S\)  in each
iteration. Given a pivot \(p \in V,\)  for each vertex \(v \in V\setminus {p},\)  the LP-KwikCluster algorithm [1]}
independently puts \(v\)  into \(S\)  with probability \((1 - x_{pv})\) . The
refined algorithm of [2]} also does independent rounding,
but puts \(v\)  into \(S\)  with probability \((1 - f^{s}(x_{pv}))\)  where \(s\in \lbrace  +, - \rbrace \)  is the sign of \((p, v)\) . (It sets \(f^{-}(x) = x\)  and
\(f^{+}(x) = 0\)  if \(x < 0.19\) , \((\frac{x-0.19}{0.5095-0.19})^2\)  if \(x \in [0.19,0.5095]\) , and 1 if \(x \ge 0.5095\) .) In order to use the power of
Sherali-Adams, given a pivot \(p\) , we round medium \(+\) edges in a
correlated manner while also employing nontrivial \(f^s(\cdot )\) 
functions for other edges. The full algorithm is described as
Algorithm REF .
